Question
If $$\alpha ,\beta \ne 0,\,{\text{and }}\,f\left( n \right) = {\alpha ^n} + {\beta ^n}$$ and \[\left| \begin{array}{l}
\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 + f\left( 1 \right)\,\,\,\,\,\,\,\,\,1 + f\left( 2 \right)\\
1 + f\left( 1 \right)\,\,\,\,\,\,\,\,\,1 + f\left( 2 \right)\,\,\,\,\,\,\,\,\,\,1 + f\left( 3 \right)\\
1 + f\left( 2 \right)\,\,\,\,\,\,\,\,1 + f\left( 3 \right)\,\,\,\,\,\,\,\,\,\,1 + f\left( 4 \right)
\end{array} \right| = K{\left( {1 - \alpha } \right)^2}{\left( {1 - \beta } \right)^2}{\left( {\alpha - \beta } \right)^2},\] then $$K$$ is equal to:
A.
1
B.
$$- 1$$
C.
$$\alpha \beta $$
D.
$$\frac{1}{{\alpha \beta }}$$
Answer :
1
Solution :
Consider
\[\left| \begin{array}{l}
\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 + f\left( 1 \right)\,\,\,\,\,\,\,\,\,\,1 + f\left( 2 \right)\\
1 + f\left( 1 \right)\,\,\,\,\,\,\,\,\,1 + f\left( 2 \right)\,\,\,\,\,\,\,\,\,\,1 + f\left( 3 \right)\\
1 + f\left( 2 \right)\,\,\,\,\,\,\,\,1 + f\left( 3 \right)\,\,\,\,\,\,\,\,\,\,1 + f\left( 4 \right)
\end{array} \right|\]
\[ = \left| \begin{array}{l}
\,\,\,1 + 1 + 1\,\,\,\,\,\,\,\,\,\,\,\,\,1 + \alpha + \beta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 + {\alpha ^2} + {\beta ^2}\\
\,\,1 + \alpha + \beta \,\,\,\,\,\,\,\,\,\,\,\,1 + {\alpha ^2} + {\beta ^2}\,\,\,\,\,\,\,\,\,1 + {\alpha ^3} + {\beta ^3}\\
1 + {\alpha ^2} + {\beta ^2}\,\,\,\,\,\,1 + {\alpha ^3} + {\beta ^3}\,\,\,\,\,\,\,\,\,\,1 + {\alpha ^4} + {\beta ^4}
\end{array} \right|\]
\[ = \left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\,\,\,\,\alpha \,\,\,\,\,\,\,\,\,\beta \\
1\,\,\,\,\,\,\,{\alpha ^2}\,\,\,\,\,\,\,{\beta ^2}
\end{array} \right| \times \,\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\,\,\,\,\alpha \,\,\,\,\,\,\,\,\,\beta \\
1\,\,\,\,\,\,\,{\alpha ^2}\,\,\,\,\,\,\,{\beta ^2}
\end{array} \right|\]
\[ = {\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\,\,\,\,\alpha \,\,\,\,\,\,\,\,\,\beta \\
1\,\,\,\,\,\,\,{\alpha ^2}\,\,\,\,\,\,\,{\beta ^2}
\end{array} \right|^2}\]
$$\eqalign{
& = {\left[ {\left( {1 - \alpha } \right)\left( {1 - \beta } \right)\left( {\alpha - \beta } \right)} \right]^2} \cr
& {\text{So, }}\boxed{k = 1} \cr} $$